Introduction
Concepts in precise set theory often have many different plausible counterparts in fuzzy set theory. In the context of choice theory, the notion of "choice set" and of "transitivity" when preferences are precise are two concepts which have many different possible interpretations in the theory of choice with fuzzy preferences. Results obtained in the literature on choice with fuzzy preferences, very often, depend significantly on the definition of choice set and of transitivity that is used (see for instance Basu (1984) Dutta et al. (1986), (henceforth, DPP (1986)), Barrett et al. (1990) Dasgupta and Deb ((1991)). This suggests a need for carefully analyzing the fuzzy counterparts of these two precise choice theoretic concepts to determine their appropriateness.
The intuitive plausibility of alternative rules for choosing ("choice functions") when preferences are fuzzy has been discussed in some detail in the literature (see Dasgupta andDeb (1991) and, in particular, Barrett, Pattanaik and Salles (1990). A similar analysis of transitivity concepts has not been done. In this paper, we evaluate eight different transitivity conditions. Three criteria are used in the evaluation process: (a) Restrictiveness (the extent to which imposition of transitivity prevents fuzzy preferences from being "truly fuzzy"); (b) Factorizability (whether the transitivity of fuzzy "weak preference" can be factored into the transitivity of "fuzzy strict preference" and the transitivity of "fuzzy indifference"); (c) Normality (the degree to which transitivity prevents choice cycles, ensuring "consistency" of choice). A tabular summary of our conclusions is provided in the last section of the paper.
1. Notations and definitions
Let X be a nonempty finite set of alternatives with the cardinality of X, I XI = n >_ 3 and ~ be the set of nonempty and nonfuzzy subsets of X. A fuzzy binary weak
The authors would like to thank a referee at Social Choice and Welfare for insightful comments.